# Monday, February 21

:::{.remark}
Recall the definitions of $\cocolim F$ and $\colim F$ for $F \in [\cat I, \cat C] = \cat{C}^{\cat I}$ with $\cat I$ a small index category.
Note that if $\cat N \da \opcat{ \Open(X)}$, the functor category $\cat{C}^{\cat N} = \Presh(X; \cat C)$ consists of presheaves on $X$.
:::

:::{.lemma title="?"}
If any of the following exist in $\cat{C}$:

- $\prod A_i$
- $\coprod A_i$
- $\cocolim F$
- $\colim F$

Then the same is true in $\cat{C}^{\cat N}$.
:::

:::{.proof title="?"}

\begin{tikzcd}
	i &&&& {F_i(U)} &&& {F_i(V)} \\
	\\
	j &&&& {F_j(U)} &&& {F_j(V)} \\
	\\
	&&& {\colim_i F_i(U)} &&& {\colim F_i(V)} \\
	\\
	& {\forall\, G(U)} &&& {\forall \,G(V)}
	\arrow[from=1-1, to=3-1]
	\arrow[from=1-5, to=3-5]
	\arrow[from=1-8, to=3-8]
	\arrow[from=3-8, to=5-7]
	\arrow[from=3-5, to=5-4]
	\arrow["{\exists !}"{description}, dashed, from=5-4, to=7-2]
	\arrow["{\exists !}"{description}, dashed, from=5-7, to=7-5]
	\arrow[from=1-5, to=5-4]
	\arrow[from=1-5, to=7-2]
	\arrow[from=1-8, to=5-7]
	\arrow[from=1-8, to=7-5]
	\arrow[curve={height=-30pt}, from=3-5, to=7-2]
	\arrow[curve={height=-30pt}, from=3-8, to=7-5]
	\arrow["{\text{claim: } \exists}", color={rgb,255:red,214;green,92;blue,92}, squiggly, from=5-4, to=5-7]
	\arrow[from=1-5, to=1-8]
	\arrow[from=3-5, to=3-8]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=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)

:::

:::{.lemma title="?"}
If $\cat{C}$ has coproducts or colimits, then so does $\Sh(X; \cat C)$.
:::

:::{.proof title="?"}
Factor through the sheafification:

\begin{tikzcd}
	{F_i} \\
	\\
	&& {(\colim F)^-} &&&& G \\
	\\
	{F_j} &&&& \textcolor{rgb,255:red,92;green,92;blue,214}{(\colim F)^{-+}}
	\arrow["{\in \Presh(X, \cat{C})}"{description}, dashed, from=1-1, to=3-3]
	\arrow["{\in \Presh(X, \cat{C})}"{description}, dashed, from=5-1, to=3-3]
	\arrow[from=1-1, to=5-1]
	\arrow[from=1-1, to=3-7]
	\arrow[from=5-1, to=3-7]
	\arrow[dashed, from=3-3, to=3-7]
	\arrow["{\exists ! (\wait)^+}", color={rgb,255:red,92;green,92;blue,214}, dotted, from=3-3, to=5-5]
	\arrow["{\exists!}", color={rgb,255:red,92;green,92;blue,214}, dotted, from=5-5, to=3-7]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=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)

:::

:::{.remark}
In $\Ab\Grp$, we have $\prod,\coprod = \bigoplus, \colim, \cocolim$.

\begin{tikzcd}
	&&&& {A_i} \\
	\\
	{\cocolim A_i} && {\prod A_i} &&&& {\coprod A_i = \bigoplus A_i} && {\colim A_i} \\
	{\ts{(a_i) \st f_{ij}(a_i) = a_j }} &&&&&&&& {\bigoplus A_i/ \gens{a_i - f_{ij}(a_i)}} \\
	&&&& {A_j}
	\arrow["{f_{ij}}", from=1-5, to=5-5]
	\arrow[squiggly, from=1-5, to=3-7]
	\arrow[squiggly, from=5-5, to=3-7]
	\arrow[squiggly, from=3-3, to=1-5]
	\arrow[squiggly, from=3-3, to=5-5]
	\arrow[from=3-1, to=1-5]
	\arrow[from=3-1, to=5-5]
	\arrow[hook, from=3-1, to=3-3]
	\arrow[from=1-5, to=3-9]
	\arrow[from=5-5, to=3-9]
	\arrow[two heads, from=3-7, to=3-9]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=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)

Note that the inner diamond doesn't necessarily commute.
The same diagram holds in $\rmod$.

:::

:::{.corollary title="?"}
In $\Sh(X, \Ab\Grp)$ and $\Sh(X, \mods{\OO_X})$, both $\oplus$ and $\colim$ exist.
:::

:::{.lemma title="?"}
In $\Sh(X, \Ab\Grp)$ and $\Sh(X, \mods{\OO_X})$, both $\prod$ and $\cocolim$ exist.
:::

:::{.proof title="?"}
In $\presh(X,\Ab\Grp)$, there exist $\prod, \cocolim$ where $(\prod F_i)(U) = \prod F_i(U)$, but this already forms a sheaf.
Check that if $U = \Union U_\alpha$, then a collection of sections $F_i(U_\alpha)$ agreeing on intersections is the same as an element of the product.
:::

:::{.warnings}
Luckily we don't need to sheafify here, since the arrow for sheafification goes the wrong way.
However, the presheaf $U \mapsto \oplus_i F_i(U)$ is not necessarily a sheaf.
Take $X = \ZZ$ with the discrete topology, then any global section has infinitely many nonzero components.
Note that $(\oplus F_i)^{-+} \subseteq \prod F_i$ is the subsheaf of the product where every local section has all but finitely many entries zero.
:::

:::{.question}
\[
\qty{\bigoplus F_i}^{-+}_p =_? \oplus (F_i)_p
,\]
i.e. is the stalk given as $\ts{ (a_i) \in (F_i)_p \st \text{ all but finitely many entries are zero}}$.
Idea: each $a_n$ might only lift to a disc of radius $1/n$, which intersect to $\ts{p}$.
For example, take $\mcf = C^\infty$ and take smooth compactly supported functions on $[-1/n, 1/n]$ converging to $\chi_{x=0}$.
:::